{
  "id": "2010.15451",
  "version": "v1",
  "published": "2020-10-29T09:49:14.000Z",
  "updated": "2020-10-29T09:49:14.000Z",
  "title": "Compactness of commutator of Riesz transforms in the two weight setting",
  "authors": [
    "Michael Lacey",
    "Ji Li"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calder\\'on--Zygmund operator $T$, and a pair of weights $ \\sigma , \\omega \\in A_p$, the commutator $ [T, b]$ is compact from $ L ^{p} (\\sigma ) \\to L ^{p} (\\omega )$ if and only if $ b \\in VMO _{\\nu }$, where $ \\nu = (\\sigma / \\omega ) ^{1/p}$. This extends the work of the first author, Holmes and Wick. The weighted $VMO$ spaces are different from the classical $ VMO$ space. In dimension $ d =1$, compactly supported and smooth functions are dense in $ VMO _{\\nu }$, but this need not hold in dimensions $ d \\geq 2$. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-29T09:49:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riesz transforms",
      "commutator",
      "weight setting",
      "compactness",
      "little vmo space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}